数字量子哈密顿量下降的编码选择与容错资源估计
Encoding Choices and Fault-Tolerant Resource Estimates for Digital Quantum Hamiltonian Descent
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中文总结 AI 辅助
研究数字量子哈密顿量下降的编码选择与容错资源估计，对比独热编码和二进制幅度编码，推导门计数缩放比例等，发现二进制编码有优势，低动量谱等可降成本，但艾克利目标中势能合成或主导成本，需利用目标函数结构降资源。
中文摘要 AI 辅助
量子哈密顿量下降（QHD）将连续优化表述为随时间变化的量子动力学，其中动能项驱动探索，势能项编码目标函数。QHD的数字实现需要将搜索空间编码到量子比特中，这种选择会改变逻辑量子比特、电路深度、非克利福德旋转和势能合成之间的主要成本。在这项工作中，我们进行了一种编码感知资源分析，比较了QHD的独热编码和二进制幅度编码。我们推导出门计数缩放比例，针对经典的薛定谔方程求解器构建并验证电路，并在基准优化问题上估计克利福德+$R_z$和容错克利福德+$T$资源。二进制编码将数据寄存器从$O(dN)$个量子比特减少到$O(d\log N)$个量子比特，并且在动能和势能演化方面给出了相当的渐近缩放比例。在所有研究的基准问题中，二进制编码使用的$R_z$旋转也比独热编码少，使其成为容错实现的首选选项，在这种实现中任意旋转占主导成本。基于低动量谱和近似量子傅里叶变换的动能近似可以进一步将二进制动能成本降低到多对数缩放比例。然而，对于像艾克利这样的目标，势能合成可能占总成本主导并降低动能近似的益处。这些结果表明，为了进一步减少资源，需要利用目标函数的解析结构来更有效地编译QHD中的势能演化。
英文摘要
Quantum Hamiltonian descent (QHD) formulates continuous optimization as time-dependent quantum dynamics, where a kinetic term drives exploration and a potential term encodes the objective function. Digital implementations of QHD require encoding the search space into qubits, and this choice can shift the dominant cost among logical qubits, circuit depth, non-Clifford rotations, and potential synthesis. In this work, we present an encoding-aware resource analysis comparing one-hot and binary amplitude encodings for QHD. We derive gate-count scalings, construct and validate circuits against classical \Sch-equation solvers, and estimate Clifford+$R_z$ and fault-tolerant Clifford+$T$ resources on benchmark optimization problems. Binary encoding reduces the data register from $O(dN)$ to $O(d\log N)$ qubits and gives comparable asymptotic scaling for both kinetic and potential evolutions. Across all benchmark problems studied, binary encoding also uses fewer $R_z$ rotations than one-hot encoding, making it the preferred option for fault-tolerant implementations where arbitrary rotations dominate the cost. Kinetic approximations based on low-momentum spectra and approximate QFTs can further reduce the binary kinetic cost to polylogarithmic scaling. However, for targets such as Ackley, potential synthesis can dominate the total cost and reduce the benefit of kinetic approximations. These results suggest that exploiting the analytic structure of the target function to compile the potential evolution in QHD more efficiently is needed for further resource reductions.
